# Systems of Linear Partial Differential Equations: Part 2

This is a brief post. Recall from Part 1 on Systems of Linear Partial Differential Equations that we wanted to find a real-analytic function $$h(x, y, \theta)$$ solving $\frac{\partial h}{\partial \theta} = 0,$ that is constant on $$x = 2$$ and zero on $$(x+1)^2 + y^2 - 1 = 0.$$ One choice would be $h(x, y) = \frac{(x + 1)^2 + y^2 - 1}{y^2 + 8}.$ The solution does not have a globally convergent Taylor series as the singularity at $$y = \pm i\sqrt{8}$$ obstructs the radius of convergence. This example demonstrates that a local multi-variable power series solution cannot be extended in the hopes of solving boundary conditions globally over $$\mathbb{R}.$$

To find this solution, we leverage the fact that $$x - 2 = 0$$ is the level set of a real analytic function. There exists a real analytic function $$g(x,y)$$ equivalent to $$(x+1)^2 + y^2 - 1$$ on $$x - 2 = 0$$ but not equal on an open set containing the vertical axis $$x - 2 = 0$$; that is the function $$y^2 + 8.$$ The problem is harder when the (system of) PDE(s) has coupling in the variables that appear in the boundary conditions.